This book provides an up-to-date description of the methods needed to face the existence of solutions to some nonlinear boundary value problems. All important and interesting aspects of the theory of periodic solutions of ordinary differential equations related to the physical and mathematical question of resonance are treated. The author has chosen as a model example the periodic problem for a second order scalar differential equation. In a paedagogical style the author takes the reader step by step from the basics to the most advanced existence results in the field.

This textbook presents all the basics for the first two years of a course in mathematical analysis, from the natural numbers to Stokes-Cartan Theorem. The main novelty which distinguishes this book is the choice of introducing the Kurzweil-Henstock integral from the very beginning. Although this approach requires a small additional effort by the student, it will be compensated by a substantial advantage in the development of the theory, and later on when learning about more advanced topics. The text guides the reader with clarity in the discovery of the many different subjects, providing all necessary tools – no preliminaries are needed. Both students and their instructors will benefit from this book and its novel approach, turning their course in mathematical analysis into a gratifying and successful experience.

This beginners' course provides students with a general and sufficiently easy to grasp theory of the Kurzweil-Henstock integral. The integral is indeed more general than Lebesgue's in RN, but its construction is rather simple, since it makes use of Riemann sums which, being geometrically viewable, are more easy to be understood. The theory is developed also for functions of several variables, and for differential forms, as well, finally leading to the celebrated Stokes-Cartan formula. In the appendices, differential calculus in RN is reviewed, with the theory of differentiable manifolds. Also, the Banach-Tarski paradox is presented here, with a complete proof, a rather peculiar argument for this type of monographs.

This book provides an up-to-date description of the methods needed to face the existence of solutions to some nonlinear boundary value problems. All important and interesting aspects of the theory of periodic solutions of ordinary differential equations related to the physical and mathematical question of resonance are treated. The author has chosen as a model example the periodic problem for a second order scalar differential equation. In a paedagogical style the author takes the reader step by step from the basics to the most advanced existence results in the field.